L(s) = 1 | − 0.758·2-s + (0.5 + 0.866i)3-s − 1.42·4-s + (−0.357 − 0.619i)5-s + (−0.379 − 0.656i)6-s + (−1.32 − 2.29i)7-s + 2.59·8-s + (−0.499 + 0.866i)9-s + (0.271 + 0.470i)10-s + (−2.24 − 3.88i)11-s + (−0.712 − 1.23i)12-s + (3.26 − 1.52i)13-s + (1.00 + 1.73i)14-s + (0.357 − 0.619i)15-s + 0.879·16-s − 3.77·17-s + ⋯ |
L(s) = 1 | − 0.536·2-s + (0.288 + 0.499i)3-s − 0.712·4-s + (−0.160 − 0.277i)5-s + (−0.154 − 0.268i)6-s + (−0.500 − 0.865i)7-s + 0.918·8-s + (−0.166 + 0.288i)9-s + (0.0858 + 0.148i)10-s + (−0.676 − 1.17i)11-s + (−0.205 − 0.356i)12-s + (0.905 − 0.424i)13-s + (0.268 + 0.464i)14-s + (0.0924 − 0.160i)15-s + 0.219·16-s − 0.915·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.477198 - 0.428687i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.477198 - 0.428687i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (1.32 + 2.29i)T \) |
| 13 | \( 1 + (-3.26 + 1.52i)T \) |
good | 2 | \( 1 + 0.758T + 2T^{2} \) |
| 5 | \( 1 + (0.357 + 0.619i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.24 + 3.88i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 3.77T + 17T^{2} \) |
| 19 | \( 1 + (-2.96 + 5.12i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 0.931T + 23T^{2} \) |
| 29 | \( 1 + (1.12 - 1.94i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.191 + 0.331i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.657T + 37T^{2} \) |
| 41 | \( 1 + (2.29 - 3.97i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.50 + 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.18 + 7.24i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.21 - 2.10i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 5.61T + 59T^{2} \) |
| 61 | \( 1 + (0.996 - 1.72i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.23 - 12.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.14 - 3.71i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.34 - 2.32i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.75 - 4.76i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + (1.79 + 3.11i)T + (-48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.26255543305204579112769405048, −10.59526768283151046386255127432, −9.755137337195632420250776915147, −8.690666606003351548687712575671, −8.254030674091836580077784637737, −6.93086766545303043066181749764, −5.41010286722957374300216835064, −4.28936696117817607146158286524, −3.21554651070689576493645361242, −0.59068248422909359583722099575,
1.89840338369393129563452682647, 3.54624152778158737587296641964, 4.98931852301615493361885258108, 6.30523584190365207664316988520, 7.46993829046625576255243907069, 8.336152512885728190326863438709, 9.243526954434769248601671729193, 9.938776122588834021568859805268, 11.11088569470695379227598092912, 12.29524269833608562713826094457